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Chapter 6: Problem 26

Divide. Write each answer in lowest terms. \(\frac{7}{8} \div \frac{3}{4}\)

Short Answer

Expert verified
\(\frac{7}{6}\)

Step by step solution

01

- Find the Reciprocal of the Divisor

To divide fractions, first find the reciprocal (or multiplicative inverse) of the divisor. For \( \frac{3}{4} \), the reciprocal is \( \frac{4}{3} \).
02

- Multiply the Numerator and the Denominator

Replace the division operation with multiplication by the reciprocal you found: \( \frac{7}{8} \times \frac{4}{3} \).
03

- Multiply Across

Multiply the numerators and multiply the denominators: \( \frac{7\times4}{8\times3} = \frac{28}{24} \).
04

- Simplify the Fraction

Simplify \( \frac{28}{24} \) by finding the greatest common divisor (GCD) of 28 and 24, which is 4. Then divide both the numerator and the denominator by the GCD: \( \frac{28 \/ 4}{24 \/ 4} = \frac{7}{6} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reciprocal of a fraction
Division of fractions might seem tricky, but it's made simpler by using the concept of the reciprocal. The reciprocal of a fraction is what you get when you flip the numerator (top number) and the denominator (bottom number). For instance, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\).

Why do we need the reciprocal? When dividing by a fraction, you multiply by its reciprocal instead. Instead of \(\frac{7}{8} \div \frac{3}{4}\), we compute \(\frac{7}{8} \times \frac{4}{3}\). This method works because dividing by a number is the same as multiplying by its reciprocal.
multiplying fractions
Once you have the reciprocal, the rest becomes straightforward multiplication. Multiplying fractions involves a simple rule: Multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Given \(\frac{7}{8} \times \frac{4}{3}\), you do the following:

  • Numerator: \(7 \times 4 = 28\)
  • Denominator: \(8 \times 3 = 24\)
This gives you \(\frac{28}{24}\). Remember to always check your multiplication.
simplifying fractions
After multiplying, you often need to simplify the result. Simplifying fractions makes them easier to understand and work with. Simplify \(\frac{28}{24}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Simplifying helps present the fraction in its lowest terms. This can be done in the following way:
  • Identify the GCD of 28 and 24, which is 4.
  • Divide both the numerator and the denominator by the GCD: \( \frac{28}{24} = \frac{28 \div 4}{24 \div 4} = \frac{7}{6}\)
Hence, \(\frac{7}{6}\) is the simplified form of \(\frac{28}{24}\).
greatest common divisor
The greatest common divisor (GCD) is crucial for simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. For our example, the GCD of 28 and 24 is 4.

Finding the GCD can be done using the following steps:
  • List the factors of each number. Factors of 28 are 1, 2, 4, 7, 14, 28. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
  • Identify the largest common factor. Here, it's 4.
This helps to reduce the fraction effectively and maintain equivalent value.

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Most popular questions from this chapter

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 5+\frac{5}{5+\frac{5}{5+5}} $$

If we write \(\frac{2 x+5}{x-4}\) as an equivalent fraction with denominator \(7 x-28,\) by what number are we actually multiplying the fraction?

Solve each formula or equation for the specified variable. $$ 9 x+\frac{3}{z}=\frac{5}{y} \text { for } z $$

Solve each formula or equation for the specified variable. $$ d=\frac{2 S}{n(a+L)} \text { for } S $$

\(\frac{-2 a}{9 a-18}=\frac{?}{18 a-36}\)
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